My support case with Apple regarding this issue got escalated to the engineering team. Sadly, I was told there is absolutely no way to transfer my passwords. Unless 1) I use iCloud or 2) had checked the encrypted check box in Time Machine.
Yeah, it makes sense that Time Machine will not move sensitive password data onto a non-encrypted directory. I am using encryption for the entire external disk - but Time Machine has no way of knowing that. I just wish there had been some type of message alerting me to this preventable problem at the time I created the backup. I am glad that I'm not being forced to use iCloud for passwords. I don't use iCloud and never plan to.
I did have an idea (workaround) - and brought this up to Apple a few times - but I'm not sure I was able to communicate that idea well enough or if Apple is aware that this idea wouldn't work.
I still have both machines. I can see every Safari password on the old machine (easy to see - just can't transfer). I could make another backup of the old machine using Time Machine with encryption turned on and then *somehow* import those passwords (and only those passwords) onto the new machine.
This seems plausible - but I'm a bit shy to try this myself without help.
Side note: While attempting to manually transfer the password files from the old machine to the new machine - I took note that it does appear that all of the password files were originally restored by Time Machine - onto the new machine. I don't know what that means though.